The Stacks project

Lemma 15.80.7. Let $R \to A \to B$ be finite type ring maps. Let $M$ be a $B$-module. If $M$ is finitely presented relative to $A$ and $A$ is of finite presentation over $R$, then $M$ is finitely presented relative to $R$.

Proof. Choose a surjection $A[x_1, \ldots , x_ n] \to B$. Choose a presentation

\[ A[x_1, \ldots , x_ n]^{\oplus s} \to A[x_1, \ldots , x_ n]^{\oplus t} \to M \to 0 \]

given by a matrix $(h_{ij})$. Choose a presentation

\[ A = R[y_1, \ldots , y_ m]/(g_1, \ldots , g_ u). \]

Choose $h'_{ij} \in R[y_1, \ldots , y_ m, x_1, \ldots , x_ n]$ mapping to $h_{ij}$. Then we obtain the presentation

\[ R[y_1, \ldots , y_ m, x_1, \ldots , x_ n]^{\oplus s + tu} \to R[y_1, \ldots , y_ m, x_1, \ldots , x_ n]^{\oplus t} \to M \to 0 \]

where the $t \times (s + tu)$-matrix is given by a first $t \times s$ block consisting of $h'_{ij}$ followed by $u$ blocks of size $t \times t$ given by $g_ iI_ t$, $i = 1, \ldots , u$. $\square$

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